Algebraic Power Sums and Newton-Girard Formulae Solutions

Problem Statement and System Analysis

• The transcript introduces a system of non-linear equations involving three variables: $a$, $b$, and $c$.
• The initial conditions provided are power sums ($s_k$) for the first three integer powers:
  • First power sum ($s_1$): $a + b + c = 4$
  • Second power sum ($s_2$): $a^2 + b^2 + c^2 = 10$
  • Third power sum ($s_3$): $a^3 + b^3 + c^3 = 22$
• The final objective, represented by the notation "a² + b + c 4 =?", is to determine the value of the fourth power sum, $s_4 = a^4 + b^4 + c^4$.

Mathematical Framework: Newton's Sums

• To solve for higher-order power sums when lower-order sums are known, one must utilize Newton's Sums (also known as the Newton-Girard formulae). These relate power sums ($s_k$) to elementary symmetric polynomials ($e_k$).
• For a set of three variables ($a$, $b$, $c$), the elementary symmetric polynomials are defined as:
  • $e_1 = a + b + c$
  • $e_2 = ab + bc + ca$
  • $e_3 = abc$
• The relationship between these polynomials and power sums is governed by the following recursive equations:
  • $s_1 - e_1 = 0$
  • $s_2 - e_1 s_1 + 2e_2 = 0$
  • $s_3 - e_1 s_2 + e_2 s_1 - 3e_3 = 0$
  • $s_n - e_1 s_{n-1} + e_2 s_{n-2} - e_3 s_{n-3} = 0$ for $n \geq 3$

Determining Elementary Symmetric Polynomials

• Step 1: Calculate $e_1$

  • From the first given equation: $s_1 = a + b + c = 4$
  • Therefore, $e_1 = 4$

• Step 2: Calculate $e_2$

  • Use the identity $s_2 = e_1^2 - 2e_2$ derived from Newton's sums.
  • Substitute the known values: $10 = (4)^2 - 2e_2$
  • $10 = 16 - 2e_2$
  • $-6 = -2e_2$
  • $e_2 = 3$

• Step 3: Calculate $e_3$

  • Use the identity for the third power sum: $s_3 = e_1 s_2 - e_2 s_1 + 3e_3$
  • Substitute the known values: $22 = (4)(10) - (3)(4) + 3e_3$
  • $22 = 40 - 12 + 3e_3$
  • $22 = 28 + 3e_3$
  • $-6 = 3e_3$
  • $e_3 = -2$

Calculating the Fourth Power Sum ($s_4$)

• To find the value for "a² + b + c 4" (interpreted as $a^4 + b^4 + c^4$), apply the general recurrence formula for $n = 4$:
  • Formula: $s_4 = e_1 s_3 - e_2 s_2 + e_3 s_1$
• Substitute the calculated elementary symmetric polynomials and the given power sums:
  • $s_4 = (4)(22) - (3)(10) + (-2)(4)$
• Perform the arithmetic operations:
  • $s_4 = 88 - 30 - 8$
  • $s_4 = 58 - 8$
  • $s_4 = 50$

Summary of Variables and Results

• The elementary symmetric polynomials for the system are:
  • $e_1 = 4$
  • $e_2 = 3$
  • $e_3 = -2$
• The corresponding power sums are:
  • $s_1 = 4$
  • $s_2 = 10$
  • $s_3 = 22$
  • $s_4 = 50$